College Algebra (10th Edition)

by
Sullivan, Michael

Answer

$y=-\sqrt{x+3}+2$

Work Step by Step

RECALL:
(i)
$y=-f(x)$ involves a reflection about the x-axis of the parent function $y=f(x)$.
(ii)
$y=f(-x)$ involves a reflection about the y-axis of the parent function $y=f(x)$.
(iii)
$y=af(x)$ involves either a vertical compression by a factor of $a$ of the parent function $f(x)$ when $a\gt 1$ or a vertical stretch when $0 \lt a \lt 1$.
(iv)
$y=f(x-h)$ involves a horizontal shift of either $h$ units to the right of the parent function $f(x)$ when $h \gt 0$ or $|h|$ units to the left when $h \lt0$.
(v)
$y=f(x) + k$ involves either a vertical shift of $k$ units upward of the parent function when $k\gt 0$ or $|k|$ units downward when $k \lt0$.
The graph involves the following transformations of the parent function $=\sqrt{x}$:
(1) Reflection about the x-axis:
Using (1) above gives:
$y=-f(x)
\\y=-\sqrt{x}$
(2) Shifted up 2 units.
Using (v) above gives:
$y=f(x)+k$ where $k=2$.
Thus, the equation is:
$\\y=-\sqrt{x}+2$
(3) Shifted to the left by $3$ units:
Using (iv) above gives:
$y=f(x-h)$ where $h=-3$.
Thus, the equation is
$\\y=-\sqrt{x-(-3)}+2
\\y=-\sqrt{x+3}+2$
Therefore, the function is:
$y=-\sqrt{x+3}+2$